Abstract
In this paper, we introduce a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. We introduce a new method of an iterative scheme {x_{n}} for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness condition and T_{omega }:= (1-omega)I+ omega T, where T is a quasi-nonexpansive mapping and omega in ( 0,frac{1}{2} ) ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result.
Highlights
Let C be a nonempty closed convex subset of a real Hilbert space H
By investigating split equilibrium problem (SEP) and SVIP, we introduce the modified split generalized equilibrium problem (MSGEP) which is to find x∗ ∈ C such that
By assuming ω is demiclosed on Motivated by SFP and SVIP, we introduced a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem
Summary
Let C be a nonempty closed convex subset of a real Hilbert space H. Many authors proved strong convergence theorem involving a quasi-nonexpansive mapping T by assuming {xn} for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without the condition above in the framework of a Hilbert space.
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